INTRODUCTION TO COMPOSITE MATERIALS

David RoylanceDepartment of Materials Science and Engineering

Massachusetts Institute of TechnologyCambridge, MA 02139

March 24, 2000

Introduction

This module introduces basic concepts of stiffness and strength underlying the mechanics offiber-reinforced advanced composite materials. This aspect of composite materials technologyis sometimes terms “micromechanics,” because it deals with the relations between macroscopicengineering properties and the microscopic distribution of the material’s constituents, namelythe volume fraction of fiber. This module will deal primarily with unidirectionally-reinforcedcontinuous-fiber composites, and with properties measured along and transverse to the fiberdirection.

Materials

The term composite could mean almost anything if taken at face value, since all materials arecomposed of dissimilar subunits if examined at close enough detail. But in modern materialsengineering, the term usually refers to a “matrix” material that is reinforced with fibers. For in-stance, the term “FRP” (for Fiber Reinforced Plastic) usually indicates a thermosetting polyestermatrix containing glass fibers, and this particular composite has the lion’s share of today’scommercial market. Figure 1 shows a laminate fabricated by “crossplying” unidirectionally-reinforced layers in a 0◦-90◦stacking sequence.Many composites used today are at the leading edge of materials technology, with perfor-

mance and costs appropriate to ultrademanding applications such as spacecraft. But heteroge-neous materials combining the best aspects of dissimilar constituents have been used by naturefor millions of years. Ancient society, imitating nature, used this approach as well: the Book ofExodus speaks of using straw to reinforce mud in brickmaking, without which the bricks wouldhave almost no strength.As seen in Table 11, the fibers used in modern composites have strengths and stiffnesses

far above those of traditional bulk materials. The high strengths of the glass fibers are due toprocessing that avoids the internal or surface flaws which normally weaken glass, and the strengthand stiffness of the polymeric aramid fiber is a consequence of the nearly perfect alignment ofthe molecular chains with the fiber axis.

1F.P. Gerstle, “Composites,” Encyclopedia of Polymer Science and Engineering, Wiley, New York, 1991. HereE is Young’s modulus, σb is breaking stress, εb is breaking strain, and ρ is density.

1

Figure 1: A crossplied FRP laminate, showing nonuniform fiber packing and microcracking(from Harris, 1986).

Table 1: Properties of Composite Reinforcing Fibers.

Material E σb εb ρ E/ρ σb/ρ cost(GPa) (GPa) (%) (Mg/m3) (MJ/kg) (MJ/kg) ($/kg)

E-glass 72.4 2.4 2.6 2.54 28.5 0.95 1.1S-glass 85.5 4.5 2.0 2.49 34.3 1.8 22–33aramid 124 3.6 2.3 1.45 86 2.5 22–33boron 400 3.5 1.0 2.45 163 1.43 330–440HS graphite 253 4.5 1.1 1.80 140 2.5 66–110HM graphite 520 2.4 0.6 1.85 281 1.3 220–660

Of course, these materials are not generally usable as fibers alone, and typically they areimpregnated by a matrix material that acts to transfer loads to the fibers, and also to pro-tect the fibers from abrasion and environmental attack. The matrix dilutes the properties tosome degree, but even so very high specific (weight-adjusted) properties are available from thesematerials. Metal and glass are available as matrix materials, but these are currently very ex-pensive and largely restricted to R&D laboratories. Polymers are much more commonly used,with unsaturated styrene-hardened polyesters having the majority of low-to-medium perfor-mance applications and epoxy or more sophisticated thermosets having the higher end of themarket. Thermoplastic matrix composites are increasingly attractive materials, with processingdifficulties being perhaps their principal limitation.

Stiffness

The fibers may be oriented randomly within the material, but it is also possible to arrange forthem to be oriented preferentially in the direction expected to have the highest stresses. Sucha material is said to be anisotropic (different properties in different directions), and control ofthe anisotropy is an important means of optimizing the material for specific applications. Ata microscopic level, the properties of these composites are determined by the orientation and

2

distribution of the fibers, as well as by the properties of the fiber and matrix materials. Thetopic known as composite micromechanics is concerned with developing estimates of the overallmaterial properties from these parameters.

Figure 2: Loading parallel to the fibers.

Consider a typical region of material of unit dimensions, containing a volume fraction Vf offibers all oriented in a single direction. The matrix volume fraction is then Vm = 1 − Vf . Thisregion can be idealized as shown in Fig. 2 by gathering all the fibers together, leaving the matrixto occupy the remaining volume — this is sometimes called the “slab model.” If a stress σ1 isapplied along the fiber direction, the fiber and matrix phases act in parallel to support the load.In these parallel connections the strains in each phase must be the same, so the strain ε1 in thefiber direction can be written as:

εf = εm = ε1

The forces in each phase must add to balance the total load on the material. Since the forces ineach phase are the phase stresses times the area (here numerically equal to the volume fraction),we have

σ1 = σfVf + σmVm = Ef ε1Vf + Emε1Vm

The stiffness in the fiber direction is found by dividing by the strain:

E1 =σ1ε1= Vf Ef + VmEm (1)

This relation is known as a rule of mixtures prediction of the overall modulus in terms of themoduli of the constituent phases and their volume fractions.If the stress is applied in the direction transverse to the fibers as depicted in Fig. 3, the slab

model can be applied with the fiber and matrix materials acting in series. In this case the stressin the fiber and matrix are equal (an idealization), but the deflections add to give the overalltransverse deflection. In this case it can be shown (see Prob. 5)

1

E2=VfEf+VmEm

(2)

Figure 4 shows the functional form of the parallel (Eqn. 1) and series (Eqn. 2) predictions forthe fiber- and transverse-direction moduli.The prediction of transverse modulus given by the series slab model (Eqn. 2) is considered

unreliable, in spite of its occasional agreement with experiment. Among other deficiencies the

3

Figure 3: Loading perpendicular to the fibers.

assumption of uniform matrix strain being untenable; both analytical and experimental studieshave shown substantial nonuniformity in the matirx strain. Figure 5 shows the photoelasticfringes in the matrix caused by the perturbing effect of the stiffer fibers. (A more completedescription of these phtoelasticity can be found in the Module on Experimental Strain Analysis,but this figure can be interpreted simply by noting that closely-spaced photoelastic fringes areindicative of large strain gradients.In more complicated composites, for instance those with fibers in more than one direction

or those having particulate or other nonfibrous reinforcements, Eqn. 1 provides an upper boundto the composite modulus, while Eqn. 2 is a lower bound (see Fig. 4). Most practical caseswill be somewhere between these two values, and the search for reasonable models for theseintermediate cases has occupied considerable attention in the composites research community.Perhaps the most popular model is an empirical one known as the Halpin-Tsai equation2, whichcan be written in the form:

E =Em[Ef + ξ(VfEf + VmEm)]

VfEm + VmEf + ξEm(3)

Here ξ is an adjustable parameter that results in series coupling for ξ = 0 and parallel averagingfor very large ξ.

Strength

Rule of mixtures estimates for strength proceed along lines similar to those for stiffness. Forinstance, consider a unidirectionally reinforced composite that is strained up to the value atwhich the fibers begin to break. Denoting this value εfb, the stress transmitted by the compositeis given by multiplying the stiffness (Eqn. 1):

σb = εfbE1 = Vfσfb + (1− Vf )σ∗

The stress σ∗ is the stress in the matrix, which is given by εfbEm. This relation is linear in Vf ,rising from σ∗ to the fiber breaking strength σfb = Ef εfb. However, this relation is not realisticat low fiber concentration, since the breaking strain of the matrix εmb is usually substantiallygreater than εfb. If the matrix had no fibers in it, it would fail at a stress σmb = Emεmb. If thefibers were considered to carry no load at all, having broken at ε = εfb and leaving the matrix

2c.f. J.C.. Halpin and J.L. Kardos, Polymer Engineering and Science, Vol. 16, May 1976, pp. 344–352.

4

Figure 4: Rule-of-mixtures predictions for longitudinal (E1) and transverse (E2) modulus, forglass-polyester composite (Ef = 73.7 MPa, Em = 4 GPa). Experimental data taken from Hull(1996).

to carry the remaining load, the strength of the composite would fall off with fiber fractionaccording to

σb = (1− Vf )σmb

Since the breaking strength actually observed in the composite is the greater of these twoexpressions, there will be a range of fiber fraction in which the composite is weakened by theaddition of fibers. These relations are depicted in Fig. 6.

References

1. Ashton, J.E., J.C. Halpin and P.H. Petit, Primer on Composite Materials: Analysis,TechnomicPress, Westport, CT, 1969.

2. , Harris, B., Engineering Composite Materials, The Institute of Metals, London, 1986.

3. Hull, D. and T.W. Clyne, An Introduction to Composites Materials, Cambridge UniversityPress, 1996.

4. Jones, R.M., Mechanics of Composite Materials, McGraw-Hill, New York, 1975.

5. Powell, P.C, Engineering with Polymers, Chapman and Hall, London, 1983.

6. Roylance, D., Mechanics of Materials, Wiley & Sons, New York, 1996.

5

Figure 5: Photoelastic (isochromatic) fringes in a composite model subjected to transversetension (from Hull, 1996).

Figure 6: Strength of unidirectional composite in fiber direction.

Problems

1. Compute the longitudinal and transverse stiffness (E1, E2) of an S-glass epoxy lamina fora fiber volume fraction Vf = 0.7, using the fiber properties from Table 1, and matrixproperties from the Module on Materials Properties.

2. Plot the longitudinal stiffness E1 of an E-glass/nylon unidirectionally-reinforced composite,as a function of the volume fraction Vf .

3. Plot the longitudinal tensile strength of a E-glass/epoxy unidirectionally-reinforced com-posite, as a function of the volume fraction Vf .

4. What is the maximum fiber volume fraction Vf that could be obtained in a unidirectionallyreinforced with optimal fiber packing?

5. Using the slab model and assuming uniform strain in the matrix, show the transversemodulus of a unidirectionally-reinforced composite to be

6

1

E2=VfEf+VmEm

or in terms of compliances

C2 = CfVf + CmVm

7